Problem 4.62
Beam
AB
has rectangular cross section where point
A
is at the origin of
the coordinate system and point
B
has the coordinates
B .1:2; 0:3; 2:4/ m
.
The vector
Er1D.1 O{C12 O|C1O
k/ m
is perpendicular to the axis
AB
of the
beam, and is parallel to the thin dimension of the cross section. The vector
Er2D.29:1 O{1:2 O|14:7 O
k/ m
is perpendicular to both the axis
AB
of the
beam and to
Er1
. The force
E
P
applied to point
B
of the beam has the direction
angles xD144ı,yD72ı, and ´D60ı.
If the beam will fail when the absolute value of the moment of
E
P
about
Er1
is
4000 Nm
, or when the absolute value of the moment of
E
P
about
Er2
is
6000 Nm, determine the magnitude of E
Pthat will cause the beam to fail.
Solution
Our strategy will be to determine the moment of
E
P
about point
A
, and then use the dot product to determine
the components of this moment in the Er1and Er2directions. Then the various fail criteria will be applied.
ErAB D.1:2 O{0:3 O|C2:4 O
k/ m;(1)
Problem 4.63
Three-wheel and four-wheel all-terrain vehicles (ATVs) are shown. For
both ATVs, the combined weight of the driver and vehicle is
WD2600 N
.
During a hard turn, the tendency for the ATV to tip is modeled by the
force
F
, which acts in the negative
y
direction. Determine the value of
F
such that the resultant moment of
F
and the weight about line
AB
is zero.
Comment on which ATV is more prone to tip during hard turns.
Note:
One
of these models of ATV is no longer manufactured because of its poor safety
record.
Statics 2e 543
Problem 4.64
An open-end wrench applies the forces
F
to the head of a bolt to produce
the moment
M
, where each force
F
is normal to the surface on which it acts.
Determine Mif FD400 lb.
Problem 4.65
Abox-end wrench applies the forces
F
to the head of a bolt to produce the
moment
M
, where each force
F
is normal to the surface on which it acts.
Determine Fif MD20 ftlb.
Solution
546 Solutions Manual
Problem 4.66
The top view of a workpiece that fits loosely in a fixture for drilling is shown.
The drill bit has two edges that apply in-plane cutting forces
F
to the workpiece.
(a)
If
FD600 N
, determine the forces
Q
between the workpiece and fixture
so that the resultant couple moment is zero when ˛D30ı.
(b)
Does your answer for
Q
from Part (a) change if
˛
has a different value?
If yes, then repeat Part (a) with ˛D60ı.
Problem 4.67
To increase the volume of a stereo, a person applies the forces
NA
,
FA
,
NB
, and
FB
to the volume control knob. All of these
forces lie in the same plane, which is parallel to the
y´
plane, and
NA
and
NB
have the same line of action, which passes through
the
x
axis,
FA
is perpendicular to
NA
, and
FB
is perpendicular
to
NB
. To prevent the knob from slipping, the person applies the
forces so that FAD0:2NAand FBD0:2NB.
(a)
If
NAD6
N and the forces applied by the person are to be
a couple or a system of couples, determine the values of
FA,NB, and FB.
(b)
Using the results of Part (a), determine the resultant couple
moment vector applied to the knob.
Solution
Problem 4.68
The large wrench shown is shaped so that a worker can use both hands to apply
forces to the wrench. If
FAD80 lb
,
˛D20ı
, and the forces at
A
and
B
are to
be a couple, determine FB,ˇ, and the moment of the couple.
Solution
Problem 4.69
Three tugboats are used to turn a barge in a narrow channel. To avoid producing
any net translation of the barge, the forces applied should be couples. The
tugboat at point Aapplies a 400 lb force.
(a) Determine FBand FCso that only couples are applied.
(b)
Using your answers to Part (a), determine the resultant couple moment
that is produced.
(c)
Resolve the forces at
A
and
B
into
x
and
y
components, and identify the
pairs of forces that constitute couples.
Problem 4.70
A bracket with pulleys for positioning magnetic tape in an electronic data
storage device is shown. The pulleys are frictionless, the tape supports a force
TD2N, and the thickness of the tape can be neglected.
(a)
Use the two tape forces in Fig. P4.70(a) to compute the resultant couple
moment.
(b)
Replace the forces on the two pulleys with point forces on the pinions
as shown in Fig. P4.70(b), and compute the resultant couple moment
(replacement of pulley forces by bearing forces is discussed in Chapter 3
in connection with Fig. 3.9).
Problem 4.71
Consider an object with forces
E
FA
and
E
FB
applied. The first column of the
following table lists resultant moments at various points due to both of these forces.
For each of the following statements, select True or False.
If E
FAand E
FBIf E
FAand E
FB
are not a couple are a couple
E
MCD ErCA E
FAC ErCB E
FBT or F? T or F?
E
MCD ErAB E
FBT or F? T or F?
E
MCD ErBA E
FAT or F? T or F?
E
MAD ErAB E
FBT or F? T or F?
E
MDD ErCA E
FAC ErCB E
FBT or F? T or F?
Problem 4.72
(a)
For loading (a) shown below, determine
FA
and
˛
so that the forces applied at points
A
and
B
are a
couple. Determine the moment of the couple.
(b) For loading (b), determine F1and F2so that this loading is equivalent to the couple in Part (a).
(c) For loading (c), determine Q1and Q2so that this loading is equivalent to the couple in Part (a).
(d) For loading (d), determine MAso that this loading is equivalent to the couple in Part (a).
Problem 4.73
Consider the object shown in Fig. P4.73(a) with forces
E
FA
,
E
FB
, and
E
FC
applied,
where points
A
,
B
, and
C
lie on the lines of action of these forces, respectively.
If
E
FACE
FBCE
FCDE
0
, show that these forces constitute a system of couples.
Hint: Consider a point
D
whose location is arbitrary. Evaluate the moment
of the three forces about point
D
and show that the result is independent of
where
D
is located, as was done in Eqs. (4.13) and (4.14) on p. 233. Using
Fig. P4.73(b), sketch a system of forces at points
A
,
B
, and
C
, including
magnitudes, that will illustrate an example of the force system described in this
problem.
Problem 4.74
To loosen the cap of a peanut butter jar requires a moment about the
´
axis of
50 in:lb
. If
E
FB
has the
x
,
y
, and
´
direction cosines
6=11
,
9=11
, and
2=11
,
respectively, and the forces
E
FA
and
E
FB
are a couple, determine the magnitude
of E
FBneeded to loosen the cap.
Problem 4.75
The manufacturer of an automobile is considering the two
designs shown for the vehicle’s lug nut wrench. Forces
FA
,
FC
,
QA
, and
QC
are applied by the user to the wrench; these
forces are perpendicular to plane
ABCD
and are to be couples.
(a)
If
FADQAD120
N, determine
FC
and
QC
, and for
each wrench, determine the moment of the couple.
(b) If a moment about the xaxis at point Dof 8000 Nmm
is required to loosen the lug nut, determine the values of
FA,FC,QA, and QC.
Problem 4.76
A structure built in at point
O
supports
300 N
and
400 N
couples. Determine the
resultant couple moment vector, using both scalar and vector approaches.
Problem 4.77
A structure built in at point
O
supports
70 N
and
85 N
couples and a tip
moment. Determine the resultant couple moment, using both scalar and
vector approaches.
Solution
From the geometry provided in the problem statement, we write vector expressions for the forces applied at
points Band Das
Problem 4.78
Determine the distance between the lines of action of the
70 N
forces.
Hint: Use the vector approach to determine the moment of this couple.
The magnitude of this result is equal to
.70
N
/d;
where
d
is the distance
between the lines of action for the two forces.
Solution
From the geometry provided in the problem statement, we write the following expressions